# Projectile: Find range, given two displacement points

## Homework Statement

A particle is projected with initial velocity $$u\cos\alpha\vec{i}+u\sin\alpha\vec{j}$$ m/s from a point 0 on a horizontal plane.

If this particle passes through two points whose displacements from 0 are $$3\vec{i}+\vec{j}$$ and $$\vec{i}+3\vec{j}$$

show that the range is $$\frac{13}{4}$$ and that $$\tan\alpha = \frac{13}{3}$$

## Homework Equations

Range

$$\frac{2u^2\sin\alpha\cos\alpha}{g}$$

S, displacement

$$S = ut+\frac{1}{2}at^2$$

## The Attempt at a Solution

So, if the initial velocity is $$u\cos\alpha\vec{i}+u\sin\alpha\vec{j}$$ m/s, then the particle is projected at u m/s at angle $$\alpha$$ to the horizontal.

Then

$$S_x = ut\cos\alpha$$
$$S_y = ut\sin\alpha-\frac{gt^2}{2}$$

and if the displacement points are $$3\vec{i}+\vec{j}$$ and $$\vec{i}+3\vec{j}$$, then:

$$S_x = ut_1\cos\alpha = 3$$
$$S_y = ut_1\sin\alpha-\frac{gt_1^2}{2} = 1$$

and

$$S_x = ut_2\cos\alpha = 1$$
$$S_y = ut_2\sin\alpha-\frac{gt_2^2}{2} = 3$$

I tried solving for the last two pair of equation. For each pair, I eliminated t. So I got two seperate equations in total involving $$\alpha$$ and u only.

So here is where I'm stuck.
I tried solving for $$\alpha$$ and u, but I just...can't!? I have no idea what I'm doing wrong, whether if it's algebraic error, or there's something wrong with the formulas. I've been stuck on this for the 5th hour now, and have redone this millions of times.. But still..

I'm beginning to suspect a problem with the question itself! Though I seriously doubt it..

Help'd be muchly appreciated!